Summary
To perform a Snap Back, perform a QCF motion with either of the two Assist Buttons. If you use the Assist 1 Button and strike the opponent with your Snap Back, you will bring in their Assist Character 1. If you use the Assist 2 Button and strike the opponent with your Snap Back, you will bring in their Assist Character 2. Snap Backs cost one Level of Hyper Meter to perform.
The Uses of Snap Backs
Landing a Snap Back will tag out your opponent's Point Character to bring in the Assist Character of your choice (depending on the button you used). Snap Backs are considered Special Moves so that you can easily combo into them from Normal Attacks. Once you've snapped out the opponent's Point Character, their new character will jump into the fray with the same behavior as if the Point Character was K.O.'ed.
Snap Backs are particularly useful for two reasons: 1) If the particular match up you are playing is bad, say you are a rushdown character and are having a troublesome time getting in on a keepaway character, Snap Backs allow you to try and bring in someone you have an easier time fighting. 2) Snap Backs do mimic K.O.'ing a character, so when the Assist Character jumps into the battle, they lose all their Red Health! So if you've beat up an Assist Character very badly so they have 90% Red Health and your opponent is smart enough to not call them out anymore to let them heal, snap that character in when you land a combo and watch all 90% of their Red Health go away!
When one of your opponent's characters is already K.O.'ed, then it doesn't matter which button you use to perform the Snap Back: either one will bring in the remaining character. If your opponent has no characters left at all, Snap Backs will just cause Soft Knock-Downs on your opponent and typically your Snap Back animation lasts long enough so you cannot land any combos on the opponent before they hit the ground and roll.
Strategy Corner
In MvC2, there was a concept known as the "Double Snap Back" that became a very important tactic in the game. Basically, if you manage to hit both the Point Character and one of their Assist Characters at the same time with one combo, you can combo into a Snap Back. What this will end up doing is launching the Point Character out of the screen, but leave the Assist Character behind! That means you have free reign over comboing the Assist Character without worrying about the opponent's Point Character protecting them. In MvC2, you could do this until the Assist Character was K.O.'ed but in Marvel Vs. Capcom 3, the Assist Characters are only vulnerable for about 4 seconds of real time before they become invincible and can escape. Thus, you need to do as much damage as you can in those 4 seconds before they get away.
Another problem is that if you ever let the Assist Character drop to the ground, they end up being in a Floored State, so only 
moves will hit them. In MvC2, every character could with something as simple as a Crouching Light Kick, so this was never a problem. But in Marvel Vs. Capcom 3, if you do not possess an move, your combo is pretty much done and the character will get up and get away. However, if you do have an move, it's very easy to continually combo the Assist four those 4 seconds. Characters like Doctor Doom can just repeatedly perform Standing Medium Attack, which s, until they become invincible. Keep in mind that Level 3's that go into a complex animation will not catch Assists, so don't try landing a Level 3 on them. Also, if you land a regular Hyper Combo on them, even if they are in the middle of getting hit by the Hyper Combo, as soon as the 4 seconds is up, they become invincible and fall to the ground, get up, and get away.
There is one major difference in Marvel Vs. Capcom 3 when you look at Double Snap Backs in MvC2: it actually matters who you snap in. In MvC2, regardless of who you tried snapping in, that character would never enter the screen until the Assist escaped or died. However, in Marvel Vs. Capcom 3, if your opponent calls in Assist Character 2 and you perform a Double Snap Back and try snapping in Assists Character 1, what will actually happen is that, while you are comboing Assist Character 2, Assist Character 1 will jump into the screen as the new Point Character! So in order to actually get the 4 free seconds of comboing the Assist, you'll actually have to know which Assist the opponent called and make sure you snap that Assist in. So if your opponent called in Assist Character 2, make sure you perform the Snap Back with the Assist 2 Button. That will allow you the freedom to beat up on the Assist until they become invincible.
Video on using SnapBacks against assists, courtesy of TenStars.